There is a sub-technique that could have helped you get a better answer for the first approach to example 1: perform a sanity check not only on the final value, but on any intermediate value you can think of.
In this example, when you estimated that there are 2500 counties, and that the average county has 300 towns with population greater than 10,000, that implies a lower bound for the total population of the US: assuming that all towns have exactly 10,000 people, that gets you a US population of 2,500x300x10,000=7,500,000,000! That’s 7.5 billion people. Of course, in real life, some people live in smaller towns, and some towns have more then 10,000 people, which makes the true implied estimate even larger.
At this point you know that either your estimate for number of counties, or your estimate for number of towns with population above 10,000 per county, or both, must decrease to get an implied population of about 300 million. This would have brought your overall estimate down to within a factor of 10.
Thanks, Luke, this was helpful!
There is a sub-technique that could have helped you get a better answer for the first approach to example 1: perform a sanity check not only on the final value, but on any intermediate value you can think of.
In this example, when you estimated that there are 2500 counties, and that the average county has 300 towns with population greater than 10,000, that implies a lower bound for the total population of the US: assuming that all towns have exactly 10,000 people, that gets you a US population of 2,500x300x10,000=7,500,000,000! That’s 7.5 billion people. Of course, in real life, some people live in smaller towns, and some towns have more then 10,000 people, which makes the true implied estimate even larger.
At this point you know that either your estimate for number of counties, or your estimate for number of towns with population above 10,000 per county, or both, must decrease to get an implied population of about 300 million. This would have brought your overall estimate down to within a factor of 10.